Finite Group Databasebeta

← #584329

Subgroups — #584329

Order 512.

3004 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2675C211yesmaximal, characteristic
2#14659C213yesmaximal, complemented
3#14651C211yesmaximal, characteristic, complemented
4#14637C211yesmaximal, characteristic, complemented
5#14623C213yesmaximal, complemented
6#14562C211yesmaximal, characteristic, complemented
7#14392C213yesmaximal, complemented
8#14363C213yesmaximal, complemented
9#14331C211yesmaximal, characteristic, complemented
10#14321C211yesmaximal, characteristic, complemented
11#14187C213yesmaximal, complemented
12#14100C213yesmaximal, complemented
13#13998C213yesmaximal, complemented
14#13983C213yesmaximal, complemented
15C4×#396C211yesmaximal, characteristic, complemented
16C22×#121C2213yes—
17C22×#121C2213yes—
18C22×C4×D4C2211yescharacteristic
19#2675C4116yescomplemented
20C2×#396C2211yescharacteristic
21C2×#396C2211yescharacteristic
22C2×#392C2213yes—
23C2×#392C2211yescharacteristic
24C2×#390C2213yes—
25C2×#390C2213yes—
26C2×#390C2213yes—
27C2×#389C2213yes—
28C2×#388C2213yes—
29C2×#388C2213yes—
30C2×#386C2213yes—
31C2×#386C2213yes—
32C2×#385C2213yes—
33C2×#381C2213yes—
34C2×#380C2213yes—
35C2×C4×C4⋊C4C2213yes—
36C2×C4×C22⋊C4C2213yes—
37C2×C4×C22⋊C4C2213yes—
38#1445C2213yes—
39#1445C2213yes—
40#1444C2211yescharacteristic
41#1443C2211yescharacteristic
42#1443C2211yescharacteristic
43#1442C2213yes—
44#1441C2213yes—
45#1441C2213yes—
46#1440C2213yes—
47#1440C2213yes—
48#1439C2213yes—
49#1439C2213yes—
50#1439C2213yes—
51#1439C2213yes—
52#1439C2213yes—
53#1439C2213yes—
54#1438C2213yes—
55#1438C2213yes—
56#1438C2213yes—
57#1438C2213yes—
58#1438C2213yes—
59#1438C2213yes—
60#1438C2213yes—
61#1438C2213yes—
62#1437C2213yes—
63#1437C2213yes—
64#1437C2213yes—
65#1437C2213yes—
66#1436C2213yes—
67#1436C2213yes—
68#1436C2213yes—
69#1436C2213yes—
70#1434C2213yes—
71C22⋊#95C2213yes—
72C22⋊#95C2213yes—
73C4×#95C2211yescharacteristic
74C23×D4C2313yes—
75C23×D4C2311yescharacteristic
76C23×D4C2313yes—
77C23×D4C2313yes—
78C24×C4C2311yescharacteristic
79C24×C4C2313yes—
80C24×C4C2311yescharacteristic
81C2×#121—46no—
82C2×#121—46no—
83C2×#121—46no—
84C2×#121—46no—
85C2×#121C2×C4124yescomplemented
86C2×#121C2×C4124yescomplemented
87C2×C4×D4—46no—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2313yes—
90C22×C4⋊C4C2313yes—
91C22×C4⋊C4C2313yes—
92C22×C4⋊C4C2313yes—
93C22×C4⋊C4C2313yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2313yes—
96C22×C22⋊C4C2313yes—
97C22×C22⋊C4C2313yes—
98C22×C22⋊C4C2313yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2313yes—
101C22×C22⋊C4C2313yes—
102C22×C22⋊C4C2313yes—
103C22×C22⋊C4C2313yes—
104C22×C22⋊C4C2313yes—
105C22×C22⋊C4C2313yes—
106C22×C22⋊C4C2313yes—
107C22×C22⋊C4C2313yes—
108C22×C22⋊C4C2313yes—
109C22×C22⋊C4C2313yes—
110C22×C22⋊C4C2313yes—
111C22×C22⋊C4C2313yes—
112C22×C42C2313yes—
113C22×C42C2313yes—
114C22×C42C2313yes—
115C22×C42C2311yescharacteristic
116#396C2×C418yescomplemented
117#396C2×C418yescomplemented
118#392C2×C4124yes—
119#392C2×C418yescomplemented
120#390—46no—
121#390—46no—
122#390—46no—
123#389—46no—
124#388—46no—
125#388—46no—
126#386—46no—
127#386—46no—
128#385C2×C4124yescomplemented
129#381—46no—
130#380—46no—
131C4×C4⋊C4—46no—
132C4×C22⋊C4—46no—
133C4×C22⋊C4—46no—
134C2×#95C2313yes—
135C2×#95C2313yes—
136C2×#95C2313yes—
137C2×#95C2313yes—
138C2×#95C2313yes—
139C2×#95C2313yes—
140C2×#95C2313yes—
141C2×#95C2313yes—
142C2×#95C2313yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2313yes—
146C2×#95C2313yes—
147C2×#95C2313yes—
148C2×#95C2313yes—
149C2×#95C2313yes—
150C2×#95C2313yes—
151C2×#95C2313yes—
152C2×#95C2313yes—
153C2×#95C2313yes—
154C2×#95C2311yescharacteristic
155C2×#95C2313yes—
156C25C2413yes—
157C25C2411yescharacteristic
158C25C2411yescharacteristic
159C22×D4—26no—
160C22×D4—26no—
161C22×D4—43no—
162C22×D4—43no—
163C22×D4—43no—
164C22×D4—43no—
165C22×D4—43no—
166C22×D4—43no—
167C22×D4—43no—
168C22×D4—43no—
169C22×D4—43no—
170C22×D4—43no—
171C22×D4—43no—
172C22×D4—43no—
173C22×D4—43no—
174C22×D4—43no—
175C22×D4—43no—
176C22×D4—43no—
177C22×D4—43no—
178C22×D4—43no—
179C22×D4C22×C4112yes—
180C22×D4C22×C4112yes—
181C22×D4C22×C4112yes—
182C22×D4C22×C414yes—
183C23×C4—26no—
184C23×C4—26no—
185C23×C4—22no—
186C23×C4—26no—
187C23×C4—22no—
188C23×C4—43no—
189C23×C4—43no—
190C23×C4—43no—
191C23×C4—43no—
192C23×C4—43no—
193C23×C4—43no—
194C23×C4—43no—
195C23×C4C22×C4112yes—
196C23×C4C2413yes—
197C23×C4C2411yescharacteristic
198C23×C4C2413yes—
199C23×C4C2413yes—
200C23×C4C2413yes—
201C23×C4C2411yescharacteristic
202C23×C4C2413yes—
203C23×C4C2411yescharacteristic
204C23×C4C2413yes—
205C23×C4C2411yescharacteristic
206C23×C4C2413yes—
207C23×C4C2411yescharacteristic
208#121—448no—
209#121—448no—
210C4×D4—416no—
211C2×C4⋊C4—26no—
212C2×C4⋊C4—26no—
213C2×C4⋊C4—26no—
214C2×C4⋊C4—26no—
215C2×C4⋊C4—26no—
216C2×C4⋊C4—43no—
217C2×C4⋊C4—43no—
218C2×C4⋊C4—43no—
219C2×C4⋊C4—43no—
220C2×C4⋊C4—43no—
221C2×C4⋊C4—43no—
222C2×C4⋊C4—43no—
223C2×C4⋊C4—43no—
224C2×C4⋊C4—43no—
225C2×C4⋊C4C22×C4112yes—
226C2×C4⋊C4C22×C4112yes—
227C2×C22⋊C4—26no—
228C2×C22⋊C4—26no—
229C2×C22⋊C4—26no—
230C2×C22⋊C4—26no—
231C2×C22⋊C4—26no—
232C2×C22⋊C4—26no—
233C2×C22⋊C4—26no—
234C2×C22⋊C4—26no—
235C2×C22⋊C4—26no—
236C2×C22⋊C4—43no—
237C2×C22⋊C4—43no—
238C2×C22⋊C4—43no—
239C2×C22⋊C4—43no—
240C2×C22⋊C4—43no—
241C2×C22⋊C4—43no—
242C2×C22⋊C4—43no—
243C2×C22⋊C4—43no—
244C2×C22⋊C4—43no—
245C2×C22⋊C4—43no—
246C2×C22⋊C4—43no—
247C2×C22⋊C4—43no—
248C2×C22⋊C4—43no—
249C2×C22⋊C4—43no—
250C2×C22⋊C4—43no—
251C2×C22⋊C4—43no—
252C2×C22⋊C4—43no—
253C2×C22⋊C4—43no—
254C2×C22⋊C4—43no—
255C2×C22⋊C4—43no—
256C2×C22⋊C4—43no—
257C2×C22⋊C4—43no—
258C2×C22⋊C4—43no—
259C2×C22⋊C4—43no—
260C2×C22⋊C4—43no—
261C2×C22⋊C4—43no—
262C2×C22⋊C4—43no—
263C2×C22⋊C4—43no—
264C2×C22⋊C4—43no—
265C2×C22⋊C4—43no—
266C2×C22⋊C4—43no—
267C2×C22⋊C4—43no—
268C2×C22⋊C4—43no—
269C2×C22⋊C4—43no—
270C2×C22⋊C4—43no—
271C2×C22⋊C4—43no—
272C2×C22⋊C4—43no—
273C2×C22⋊C4C22×C4112yes—
274C2×C22⋊C4C22×C4112yes—
275C2×C22⋊C4C22×C4112yes—
276C2×C22⋊C4C22×C4112yes—
277C2×C42—26no—
278C2×C42—26no—
279C2×C42—26no—
280C2×C42—26no—
281C2×C42—43no—
282C2×C42—43no—
283C2×C42—43no—
284C2×C42—43no—
285C2×C42—43no—
286C2×C42—43no—
287#95—26no—
288#95—26no—
289#95—26no—
290#95—26no—
291#95—26no—
292#95—43no—
293#95—43no—
294#95—43no—
295#95—43no—
296#95—43no—
297#95—43no—
298#95—43no—
299#95—43no—
300#95—43no—
301#95—43no—
302#95—43no—
303#95—43no—
304#95—43no—
305#95—43no—
306#95—43no—
307#95C22×C4112yes—
308#95C22×C414yes—
309C24—23no—
310C24—23no—
311C24—23no—
312C24—23no—
313C24—23no—
314C24—23no—
315C24—23no—
316C24—23no—
317C24—23no—
318C24—23no—
319C24—23no—
320C24—23no—
321C24—23no—
322C24—23no—
323C24—23no—
324C24—23no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—23no—
329C24—21no—
330C24—23no—
331C24—23no—
332C24—23no—
333C24—23no—
334C24—23no—
335C24C23×C412yes—
336C24C23×C412yes—
337C24C23×C416yes—
338C24C2511yescharacteristic
339C2×D4—224no—
340C2×D4—412no—
341C2×D4—412no—
342C2×D4—412no—
343C2×D4—412no—
344C2×D4—412no—
345C2×D4—412no—
346C2×D4—412no—
347C2×D4—412no—
348C2×D4—412no—
349C2×D4—44no—
350C2×D4—412no—
351C2×D4—86no—
352C2×D4—86no—
353C2×D4—86no—
354C2×D4—86no—
355C2×D4—86no—
356C2×D4—86no—
357C2×D4—86no—
358C2×D4—86no—
359C2×D4—86no—
360C2×D4—86no—
361C2×D4—86no—
362C2×D4—86no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—23no—
366C22×C4—23no—
367C22×C4—21no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4—23no—
371C22×C4—23no—
372C22×C4—23no—
373C22×C4—23no—
374C22×C4—23no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—23no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—23no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4—23no—
384C22×C4—23no—
385C22×C4—224no—
386C22×C4—23no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—23no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—23no—
407C22×C4—23no—
408C22×C4—23no—
409C22×C4—21no—
410C22×C4—23no—
411C22×C4—23no—
412C22×C4—23no—
413C22×C4—23no—
414C22×C4—23no—
415C22×C4—23no—
416C22×C4—23no—
417C22×C4—23no—
418C22×C4—23no—
419C22×C4—21no—
420C22×C4—412no—
421C22×C4—412no—
422C22×C4—412no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—86no—
426C22×C4—86no—
427C22×C4—86no—
428C22×C4—86no—
429C22×C4—86no—
430C22×C4—86no—
431C22×C4—86no—
432C22×C4C23×C412yes—
433C22×C4C23×C416yes—
434C22×C4C23×C416yes—
435C22×C4C23×C416yes—
436C22×C4H(5,𝔽2)16yes—
437C22×C4H(5,𝔽2)16yes—
438C22×C4H(5,𝔽2)16yes—
439C22×C42−1+412yes—
440C4⋊C4—224no—
441C4⋊C4—412no—
442C4⋊C4—412no—
443C4⋊C4—44no—
444C4⋊C4—86no—
445C4⋊C4—86no—
446C22⋊C4—224no—
447C22⋊C4—224no—
448C22⋊C4—412no—
449C22⋊C4—412no—
450C22⋊C4—412no—
451C22⋊C4—412no—
452C22⋊C4—44no—
453C22⋊C4—412no—
454C22⋊C4—44no—
455C22⋊C4—86no—
456C22⋊C4—86no—
457C22⋊C4—86no—
458C22⋊C4—86no—
459C22⋊C4—86no—
460C22⋊C4—86no—
461C22⋊C4—86no—
462C22⋊C4—86no—
463C22⋊C4—86no—
464C42—412no—
465C42—44no—
466C42—86no—
467C42—86no—
468C23—26no—
469C23—26no—
470C23—26no—
471C23—26no—
472C23—26no—
473C23—26no—
474C23—26no—
475C23—26no—
476C23—26no—
477C23—26no—
478C23—26no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—43no—
484C23—43no—
485C23—43no—
486C23—43no—
487C23—43no—
488C23—43no—
489C23—43no—
490C23—43no—
491C23—43no—
492C23—43no—
493C23—43no—
494C23—43no—
495C23—43no—
496C23—43no—
497C23—43no—
498C23—43no—
499C23—43no—
500C23—43no—
501C23—43no—
502C23—43no—
503C23—43no—
504C23—43no—
505C23—43no—
506C23—43no—
507C23—43no—
508C23—43no—
509C23—43no—
510C23—43no—
511C23—43no—
512C23—43no—
513C23—43no—
514C23—43no—
515C23—43no—
516C23—43no—
517C23—43no—
518C23—43no—
519C23—43no—
520C23—43no—
521C23—43no—
522C23—43no—
523C23—43no—
524C23—43no—
525C23—43no—
526C23—43no—
527C23—43no—
528C23—43no—
529C23C24×C411yescharacteristic
530C23C2×H(5,𝔽2)13yes—
531C23C2×H(5,𝔽2)13yes—
532C23C2×H(5,𝔽2)13yes—
533C23C2×2−1+411yescharacteristic
534C23#58513yes—
535C23#58511yescharacteristic
536D4—416no—
537D4—824no—
538D4—824no—
539D4—824no—
540C2×C4—26no—
541C2×C4—26no—
542C2×C4—26no—
543C2×C4—26no—
544C2×C4—26no—
545C2×C4—26no—
546C2×C4—26no—
547C2×C4—26no—
548C2×C4—26no—
549C2×C4—26no—
550C2×C4—26no—
551C2×C4—26no—
552C2×C4—26no—
553C2×C4—26no—
554C2×C4—26no—
555C2×C4—26no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—43no—
559C2×C4—43no—
560C2×C4—43no—
561C2×C4—43no—
562C2×C4—43no—
563C2×C4—43no—
564C2×C4—43no—
565C2×C4—43no—
566C2×C4—43no—
567C2×C4—43no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—43no—
571C2×C4—43no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—43no—
575C2×C4—43no—
576C2×C4—43no—
577C2×C4—43no—
578C2×C4—43no—
579C2×C4—43no—
580C2×C4—43no—
581C2×C4—43no—
582C2×C4—43no—
583C2×C4—43no—
584C2×C4—43no—
585C2×C4—43no—
586C2×C4—43no—
587C2×C4—43no—
588C2×C4—43no—
589C2×C4—43no—
590C2×C4—43no—
591C2×C4—43no—
592C2×C4—43no—
593C2×C4—43no—
594C2×C4—43no—
595C2×C4—43no—
596C2×C4—43no—
597C2×C4—43no—
598C2×C4—43no—
599C2×C4—43no—
600C2×C4—43no—
601C2×C4—43no—
602C2×C4—824no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4#518112yes—
606C2×C4#56014yes—
607C22—42no—
608C22—42no—
609C22—46no—
610C22—46no—
611C22—46no—
612C22—46no—
613C22—46no—
614C22—46no—
615C22—46no—
616C22—46no—
617C22—46no—
618C22—46no—
619C22—46no—
620C22—83no—
621C22—83no—
622C22—83no—
623C22—83no—
624C22—83no—
625C22—81no—
626C22—81no—
627C22—83no—
628C22—81no—
629C22—81no—
630C22—83no—
631C22—83no—
632C22—83no—
633C22—83no—
634C22—83no—
635C22—83no—
636C22C2×#51813yes—
637C22C4×H(5,𝔽2)13yes—
638C22C4×2−1+411yescharacteristic
639C22C2×#56011yescharacteristic
640C22#352913yes—
641C22#353613yes—
642C22#353713yes—
643C22#353913yes—
644C22#354213yes—
645C22#354313yes—
646C22#354613yes—
647C22#354713yes—
648C22#355813yes—
649C4—212no—
650C4—24no—
651C4—42no—
652C4—46no—
653C4—46no—
654C4—42no—
655C4—83no—
656C4—81no—
657C4—83no—
658C4—81no—
659C4—83no—
660C4—83no—
661C2—82no—
662C2—82no—
663C2—86no—
664C2C4×#56011yescharacteristic
665C2#3441013yes—
666C2#3447113yes—
667C2#5247211yescharacteristic
668C2#5247613yes—
669C2#5255313yes—
670C2#5309011yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC2×#2675C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14659C2semidirect
3. #14651C2semidirect
4. #14637C2semidirect
5. #14623C2semidirect
6. #14562C2semidirect
7. #14392C2semidirect
8. #14363C2semidirect
9. #14331C2semidirect
10. #14321C2semidirect
11. #14187C2semidirect
12. #14100C2semidirect
13. #13998C2semidirect
14. #13983C2semidirect
15. C4×#396C2semidirect
19. #2675C4semidirect
85. C2×#121C2×C4semidirect
86. C2×#121C2×C4semidirect
116. #396C2×C4semidirect
117. #396C2×C4semidirect
119. #392C2×C4semidirect
128. #385C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#26752
#146592
#146512
#146372
#146232
#145622
#143922
#143632
#143312
#143212
#141872
#141002
#139982
#139832
C4×#3962